String Theory and Partition Numbers

Recently there was an intriguing breakthrough in the study of partition numbers. partitions numbers P(n) count the number of ways of expressing n as a sum of positive integers. E.g. P(4) = 5 because 4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 = 4. The sequence of partition numbers goes 1,1,2,3,5,7,11,15,22, … It starts off wanting to be the Fibonacci numbers but then after the first five it changes its mind and gives the first five prime numbers before degenerating into a rapidly increasing sequence of interesting numbers.

There are lots of well known things that can be said about the partition numbers but I’ll leave that to Ken Ono who is the main figure in the new discovery, actually two discoveries. The first discovery is a new finite formula for partition functions and the second is a new explanation for some congruence relations discovered by Rananujan. I highly recommend this low level lecture by Ken Ono as an introduction to the new finds.

One thing that is not mentioned in all the recent news coverage is the important connection between partition numbers and string theory that is very easy to see even a very basic level. From the theory of musical harmonics you know that a string has vibration modes labelled by integers k, whose frequency is ωk = kα for some constant α that depends on things like the tension in the string. When a string is quantized it can be treated like a set of decoupled harmonic oscillators with energy levels Ek = (1/2 + mk)ħωk where the sequnece of non-negative integers mk labels the eignestates of the oscillators. So the total energy is given by

E = Σk (1/2 + mk)ħkα

The zero-point energy is E0 = 1/2 ħkα(1+2+3+4+…) . We can either ignore this as an irrelevant constant while pretending not to notice that it is infinite, or we can use zeta regulation to deduce that 1+2+3+4+ … = ζ(-1) = -1/12. In any case, what we are really interested in is the rest of the sum and to understand it we just need a simple trick. Write mk k = ( k+k+…+k ) (mk times) Then you will immediately notice that the number of states with an energy En = E0 + nħα is exactly P(n), the partition number of n. This is also valid for the relativistic bosonic string in 26 dimensional spacetime, except that then you need to multiply by 24 because the one dimensional string can vibrate independently in any of the 24 space dimensions transverse to the string.

The partition function for bosonic string theory is therefore given by

Z = ΣnP(n) exp( – (24n-1)ħα)

Perhaps that’s why they call it the partition function :)

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This entry was posted on Saturday, February 12th, 2011 at 10:34 am and is filed under Number Theory, String Theory. You can follow any responses to this entry through the RSS 2.0 feed.
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11 Responses to String Theory and Partition Numbers

I have been busting my ass on this stuff for the last three weeks. This gets really deep, and it also makes contact with the elliptic curve theory in your paper. The partition function for the bosonic string is a Dedekind η-function, which is usually approximated so it has a form of the Hardy-Ramanujan function. However, if one considers it under various the discrete group actions the resulting generator function is a product of Eisenstein functions. The Eisenstein function is has eigenvalue -2 with the euclideanized hyerpbolic Laplacian –y(∂^2/∂x^2 + ∂^2/∂y^2). This function, F(z) under the action of (1/2π)(-i∂/∂z + 1/y] is an anti-holomorphic function that is an integer partition.

I did see the lecture, and thanks for providing it! One gets the sense that much more than a result in recreational mathematics has been achieved. In particular, the fact that their results in some sense explain the oddities of the Ramanujan identies is realy impressive. My guess is that this might just be the beginning.

Some mathematicians have said that this is merely an “unsurprising” application of the theory of p-adic modular forms and the real breakthrough is in that field. Perhaps we should expect some developments in p-adic string theory next.

I like the way that different small prime numbers have different uses. The way the partition congruences works turns out to depend on the value of [(p-1)/12] – [(p^2-1)/24p] where the square brackets are for the floor function. For 5,7 and 11 this is zero which is why there are many congruences for these primes. For primes up to 31 it equals one, so there are still congruences but they are harder to find. 2 and 3 dont give congruences because they are factors of 24.

There are similar roles for special primes in elliptic curve theory and group theory. For example the supersingular primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 are important to moonshine theory as factors of the size of the monster group and as special cases for elliptic curves modulo p. 5,7 and 11 also have special significance because PSL(2,p) is “exceptional” for these primes. In all cases it is the fact that p+1 has small factors that seems to be significant so there could be connections.

There is a lot which can impact sporadic groups as well. These groups have a lot of θ-function realizations. I am not sure what role the discrete 3-fold system from the Γ(6) 2-3-6 fold quotient system would work for the Leech lattice Λ_{24}. It may require looking into research on Eisenstein functions with sporadic groups, and E_8 as well.

Hyperbolic geometry: the metric of Minkowski space-time. Here R is the speed of light, and Equation (1.3) defines a light cone. Our original 2-sphere is the base of the light cone, the \celestial sphere” of an observer.
p.17. The classical analogy is that between the ring of integers in number theory and the polynomial rings in geometry. A \half-way house” is an algebraic curve over a finite field. A finite field is a field Fq with q elements, where q = pn for some prime p.

This math is not my strong side :) Can a ‘ring’ be an hexamer, or a ribos, a pentamer?